Ultimate limit state

The stochastic nature of variables is reflected in mean, standard deviation and type of distribution of relevant variables. An important notion is that most service-life approaches calculate the mean service life, that is the service life to be achieved with 50% probability. From the point of view of economy and safety, this is not acceptable. Depending on the severity of the adverse event occurring limit state), the failure probability should be (much) lower than 50%. Limit states can for example be the structure needs repair because concrete parts are falling off due to corrosion, or the structure collapses. The need for repair is termed a serviceability limit state (SLS). Collapse is termed an ultimate limit state (ULS). Serviceability limit states should have a low failure probability of the order of 1 100. Ultimate limit states involve safety (human lives) or loss of the structure (high economic damage) and must have a very low failure probability of the order of 1 10000. Failure probabilities like this are defined by e. g. EN 1990, Annex B and C [22]. As initiation of corrosion does not immediately have extreme consequences, a probability of failure has been proposed for this event of 1 10 [23]. 

A. Vrouwenvelder, P. Schiessl, Durability aspects of probabilistic ultimate limit state design . Heron, 1999, 44 (1), 19-30. 

Ductile properties such as crack pattern and deformations prefiguring the nearing failure are important characteristics regarding the fracture behavior of structural concrete members. The tests demonstrated that in general TRC members have a distinctive ductile behavior although the stress-strain-behavior of the fabrics is linear-elastic until a brittle tensile failure. While the deformations under service loads (SLS) are rather small, the load-bearing behavior of the specimens is characterized by a distinctive stabilized crack pattern as well as high deformations in ultimate limit state (ULS) of L/30 - L/20. 

For the limit state method instead of nominal loads which are a mixture of standard weights of material (which are defined as fair average values in B.S. 648 1964) and extreme estimates of imposed loads, 5% characteristic loads are used with various partial factors according to the nature of the loading. The design bending moments for the ultimate limit state using the partial factors of 1.4 for dead loads and 1.6 for live loads are thus. 

This ultimate moment of resistance is greater than the design moment of 378 kN m and the section is, therefore, satisfactory for this ultimate limit state. However, it has to be checked for the stresses in the serviceability limit state. As for the ultimate load method, the second moment of area of the composite section is 7.01 x 10 mm and the elastic modulus of the steel beam alone is 949 cm , and thus the stage 1 steel beam stress is... 

It is clear that with the assumptions made the probability of failure is very low. For a normal distribution a value of /Iuk of 7.2 corresponds to a probability of failure of 3.8 x 10", and for = 4.2, the figure is about 10". These are such low figures that slight differences in P cause relatively large differences in the probability of failure. A cumulative distribution function for the distribution of Z in the ultimate limit state was plotted from a histogram generated by the Monte Carlo process and showed an approximately normal distribution with a slight tendency to deviate from this at the tail (Fig. 5.10). 

The ultimate limit state design bending moment is 378 kN m but it is important to recognise which of the factors y, and yc, that the partial factor values used, represent. According to CIRIA Report 63, the values recommended in C.P. 110 include an allowance for all these factors except which is intended to take account of the nature of the structure and its behaviour. In this problem y will also be taken as unity and so the design bending moment for the ultimate limit state remains at 378 kN m. 

Fig. 4.7 Composite Section for Limit State Method (Ultimate Limit State).

For the ultimate limit state the basic variables considered are listed in Table 5.1. Considering the equations in Section 4.5 and including a system variable P then... 

Load factors as per BS8006 Table 27 for Ultimate Limit State Conditions are ... 

Reliability can he defined as the probabilistic measure of assurance of performance with respect to some prescribed conditions (Krahk,J. 2009b). A condition can refer to an ultimate limit state (such as collapse) or serviceability limit state (such as excessive deflection and/or vibration). 

It is verified by an inspection that the bottom reinforcement consists of 7 bars of the diameter 25 mm and the reinforcement area is 3.44 x 10 m /m. Due to corrosion, the area is reduced to 3.26 x 10 m /m. In general three ultimate limit states of moment, shear and combined moment and shear are assumed to be important for deteriorating reinforced concrete bridges, Sarveswaran Roberts (1999). 

In the case of deterministic calculation and the ultimate limit state (ULM) of the structure the load combination is considered according to ENV 1990 as follows ... 

The Backgroimd document (1999) provides additional information about experimental measurements of temperatures which formed the basis for the development of the models of thermal actions given in Eurocodes. The characteristic values of temperature components are based on the fifty years return period like other climatic actions (snow, wind velocity, icing). Presently, the Eurocodes recommend a unique value of partial factor yg = 1,5 for most variable actions Q with respect to the ultimate limit states. However, the reduced factor yr = 1, 2 may be applied for thermal actions in some national standards, CSN 73 6203 (1986). It appears that the partial factors for some variable actions might be differentiated taking into account their characteristics. 

It appears that the application of the unique value of the partial factor for most variable actions in the ultimate limit states should be reconsidered during the period of the Eurocodes maintenance. It may be mentioned here that some national standards... 

In the next study case it is considered that the bridge fulfills the requirements of Eurocodes for the ultimate limit states and the minimum area of reinforcement needed for limiting cracking, ft is assumed here that the inihal crack width may shghtly overcross the crack width limit (0,3 mm). The inihal... 

In general, FRP composites exhibit little or no ductile behaviour beyond a point of linear stress-strain behaviour of the material failure of the material may occur locally soon after this point has been reached. The design shonld take acconnt of this behaviour by ensuring that a serviceability limit state is reached prior to its ultimate limit state for the mode of failure being considered. 

P(2) Ultimate limit states are those associated with coUapse, or with other forms of structural failure which may endanger the safety of people or the continued performance of associated structnres or components. 

P(3) States prior to structural collapse which, for simplicity, are considered in place of the collapse itself are also treated as ultimate limit states. 

Ultimate limit states which may require consideration inclnde ... 

Partial safety factors for ultimate limit states... 

Depending on the type of structure, its function or the method of construction, design may be carried out primarily for the serviceahility limit state, i.e. maximum strain. In many cases, provided that checks for this limit state have been carried out, checks for the ultimate limit state may be dispensed with as compliance can be seen by inspection. 

P(8) Design of connections shall always be checked at the ultimate limit state. 

P(l) In the ultimate limit state, consideration shall be given to the effects of possible imperfections in the geometry of the unloaded structure. Where significant, any possible unfavourable effects of such imperfections shall be taken into account. 

P(2) For the purposes of design of the said elements at the ultimate limit state, the section properties shall be those corresponding to the nominal dimensions less the manufacmring tolerances given in Table 6.1. 

P(l) Depending on the specific nature of the structure, the limit state being considered and the specific conditions of design or execution, for the ultimate limit states a linear elastic or a second-order analysis method shall be used. 

At the ultimate limit state the effects of creep (e.g. creep rupture) shall be taken into account by the choice of the appropriate value of yn, 3 (see 2.3.3.2). 

Otherwise, provided stresses are kept within the limits corresponding to normal service conditions, the effects of creep and shrinkage need be taken into account only for the serviceability limit states, except where their influence on second order effects at the ultimate limit state is likeiy to be significant (see 2.5.1.4). 

Design resistance (usually ultimate limit state, ULS)... 

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